\section{1.10} 
\begin{frame}[allowframebreaks]{1.10. }

\vspace{-0.4cm}

{\color{red}1.10 THEOREM.}
\begin{enumerate}
\item[(i)] The sheaf of rings $\mathcal{D}_X$ is locally noetherian and coherent.
\item[(ii)] The global homological dimension of $\mathcal{D}_{X,x}$ is equal to $\dim X$ ($x \in X$), that of $\mathcal{D}_X$ is $\le 2 \cdot \dim X$.
\item[(iii)] $d_x(M) \ge \dim X$ if $M_x \ne 0$.
\item[(iv)] $d_x(M) + j_x(M) = 2 \cdot \dim X, \quad (x \in X)$
\end{enumerate}

Proof. 

(i) follows from 1.9 as in the case of the Weyl algebra, and (iv) from 1.9 and V.2.2.2. 

Let $M$ be a $\mathcal{D}$-module and fix $x \in X$. 

As recalled in 1.3 we may find an affine neighborhood $Y$ of $x$, a hyper-surface $T \subset Y$, an étale morphism $\sigma: Y - T \to \mathbb{A}^n$ ($n = \dim X$) whose image is the complement of a hypersurface $S$. 

Let $U = Y - T$ and $V = \mathbb{A}^n - S$. 

The coordinate ring $\mathcal{O}(U)$ is finitely generated, integral over $\mathcal{O}(V)$. 

We may identify $\mathcal{D}(U)$ to the subring of $\mathcal{D}(U)$ generated over $\mathcal{O}(V)$ by the lifts of the canonical derivations $\partial_i \in \mathbb{A}_n$, therefore
\begin{equation}
\mathcal{D}(U) = \mathcal{O}(U) \otimes_{\mathcal{O}(V)} \mathcal{D}(V)
\end{equation}

The $\mathcal{D}(U)$-module $M(U)$ may then be viewed as a finitely generated $\mathcal{D}(V)$-module. 

According to Thm 1, p.17 in [Bu 2]
\begin{equation}
d_{\mathcal{D}(U)}(M(U)) = d_{\mathcal{D}(V)}(M(U))
\end{equation}
where $d_{\mathcal{D}(U)}$ (resp. $d_{\mathcal{D}(V)}$) refers to the dimension of the characteristic variety of $M(U)$, viewed as a $\mathcal{D}(U)$-module (resp. $\mathcal{D}(V)$)-module.

We have $\mathcal{O}(V) = \mathcal{O}(\mathbb{A}^n)[f^{-1}]$, where $f$ generates the ideal of $S$, whence also $\mathcal{D}(V) = \mathcal{D}(\mathbb{A}^n)[f^{-1}] = \mathbb{A}_n[f^{-1}]$. 

It is standard that
\begin{equation}
\mathrm{gl}\,\mathrm{hd}(\mathbb{A}_n[f^{-1}]) \le \mathrm{gl}\,\mathrm{hd}\,\mathbb{A}_n = n
\end{equation}
therefore
\begin{equation}
j_{\mathcal{D}(V)}(M(U)) \ge n
\end{equation}
and, by (iv),
\begin{equation}
d_{\mathcal{D}(V)}(M(U)) \ge n
\end{equation}

Together with (2), this yields (iii). 

Using (iv) again, we now have
\begin{equation}
\mathrm{Ext}^j_{\mathcal{D}(U)}(M(U), \mathcal{D}(U)) = 0 \quad \text{for } j > n
\end{equation}
for every finitely generated $M(U)$. 

By V.2.3.3, the relation (6) remains true if the second argument is replaced by a finitely generated $\mathcal{D}(U)$-module and then, in view of §8, No.3 in [Bu 1], by a general $\mathcal{D}(U)$-module. 

It follows that we now have
\begin{equation}
(\mathrm{Ext}^j_{\mathcal{D}_X}(M,N))_x = 0 \quad \text{for } j > n \ (M,N \in u(\mathcal{D}_x), x \in X).
\end{equation}

Similarly, we have
\begin{equation}
\mathrm{Ext}^j_{\mathcal{D}_{X,x}}(M_x, N_x) = 0 \quad \text{for } j > n,
\end{equation}
which yields the first part of (ii). 

There exists a spectral sequence abutting to $\mathrm{Ext}^*_{\mathcal{D}_X}(M,N)$ in which
\begin{equation}
E_2^{p,q} = H^p(X, \mathrm{Ext}^q_{\mathcal{D}_X}(M,N))
\end{equation}
[Go:II,7.3.3]. 

By (7), $E_2^{q,p}$ is zero if $q > n$. 

On the other hand, the cohomology of $X$ with coefficients in a sheaf is zero above $n$ by the Serre-Grothendieck vanishing theorem [Ha 2:I,2.7], therefore
\[
\mathrm{Ext}^s_{\mathcal{D}_X}(M,N) = 0 \quad \text{for } s > 2n.
\]
This proves the second part of (ii).

\end{frame}

